Problem
Plan the most efficient route
Eight places A to H are joined by roads of known length. Start at H, visit every other place once, and return to H. Only roads on the list may be used. Find the shortest round trip and its total distance.
Reasoning · Grade 2-2 Working with Short Lengths
Your answer
Watch out Grabbing the shortest road each time strands a place and the loop cannot close — start from the fact that every place needs two roads.
How to solve
Strategy Make a Systematic List — A round trip visiting every place is built one road at a time, so we list candidate routes in an orderly way and add up their lengths. We favor the cheap roads (the many 15-20 m links) and avoid the expensive ones (B-E 50, A-B 45, C-E 40) — guess-and-check on a drawn map quickly steers us to the best route.
1STEP 1
Spot the cheap and expensive roads
Most useful roads are 15-20 m; the 40, 45 and 50 m ones are to avoid.
cheap: 15,20,… costly: 50,45,40
To keep a total small you reach for the small numbers first — same idea as packing a bag with the lightest items.
2.MD.B.5Draw A Diagram2STEP 2
Build a route that avoids the worst roads
Avoiding the pricey roads forces the loop H-E-G-A-D-B-C-F-H.
H→ E→ G→ A→ D→ B→ C→ F→ H
Each town has only a couple of short exits, so following the cheapest exits links the towns into one neat loop.
2.MD.B.5Make A Systematic List3STEP 3
Add up the chosen roads
The eight chosen roads add to 175 m.
15+20+30+20+20+30+20+20 = 175
Adding a handful of two-digit lengths (within 1000) is plain Grade 2 addition.
2.NBT.B.7Make A Systematic List4STEP 4
Check no cheaper loop exists
Any swap drags in a pricey road, so no shorter loop exists.
175 m is minimal
When every other choice forces a bigger road in, you know you have found the smallest total.
2.NBT.B.7Guess And CheckAny other loop is forced to take in a longer road, so the loop that was built really is the shortest.
Why?
If every alternative swaps a chosen road for a longer one, each alternative total is bigger, and those comparisons chain into one ranking.
🧱Transitivity of orderIf the first is smaller than the second and the second smaller than the third, the first is smaller than the third, so pairwise comparisons chain into one full order.
Why?
Once every competing loop has been ruled out for a definite reason, the one still standing is the answer.
🧱Rule out the rest and the last one is forcedIf every possibility but one has been ruled out, the survivor must be the answer — no further checking is needed.
Answer
175 m (H-E-G-A-D-B-C-F-H)
The loop uses eight roads. Their average length is about 22 m, and 8 × 22 ≈ 176, right at our total of 175 m. It also avoids all three of the most expensive roads (40, 45, 50 m), which is exactly what a shortest tour should do.
Takeaway
Pick the shortest roads first and the loop almost builds itself — then it is just Grade 2 adding to get 175 m!
- Spot the cheap and expensive roads
- Build a route that avoids the worst roads
- Add up the chosen roads
- Check no cheaper loop exists